Scaling Zernike coefficients to smaller pupil sizes for refractive treatments

ABSTRACT

Wavefront measurements of eyes are normally taken when the pupil is relatively large, and the results are often represented by a set of Zernike coefficients. Different sets of Zernike coefficients can be calculated to represent aberrations at smaller pupil sizes. While recently described techniques allow scaling of the expansion coefficients with Zernike polynomials, a more intuitive approach would be desirable. Such an approach may optionally derive an equivalent result as known techniques, but may employ a much simpler and nonrecursive formula between the new and the original sets of Zernike polynomial expansion coefficients of a wavefront when the aperture size is scaled.

CROSS REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of U.S. Provisional Application No.60/776,289, filed Feb. 24, 2006, the disclosure of which is incorporatedherein by reference.

BACKGROUND OF THE INVENTION

A recent recursive analytical formula has been derived to calculate aset of new Zernike polynomial expansion coefficients from an originalset when the size of the aperture is reduced. There may be benefits to amore intuitive derivation of a simpler, nonrecursive formula, which canbe used to calculate the instantaneous refractive power.

BRIEF SUMMARY OF THE INVENTION

The ocular aberrations of human eyes can now be measured objectivelythanks to the introduction of wavefront technology to ophthalmology. Inthe various fields of optics, wavefront aberrations have traditionallybeen represented by Zernike polynomials.

Wavefront measurements of eyes are normally taken when the pupil isrelatively large, and the results are often represented by a set ofZernike coefficients. Different sets of Zernike coefficients can becalculated to represent aberrations at smaller pupil sizes. Whilerecently described techniques allow scaling of the expansioncoefficients with Zernike polynomials, a more intuitive approach wouldbe desirable. Such an approach may optionally derive an equivalentresult as known techniques, but may employ a much simpler andnonrecursive formula between the new and the original sets of Zernikepolynomial expansion coefficients of a wavefront when the aperture sizeis scaled.

In a first aspect, embodiments of the present invention provide a methodof calculating a modified normalized Zernike expansion coefficient foran optical system. The method may include inputting an originalnormalized Zernike expansion coefficient for the optical system, wherethe original normalized Zernike expansion coefficient is associated witha first aperture dimension. The method may also include calculating amodified normalized Zernike expansion coefficient for the opticalsystem, where the modified normalized Zernike expansion coefficient isassociated with a second aperture dimension. In some cases, the modifiednormalized Zernike expansion coefficient can be calculated based on theoriginal normalized Zernike expansion coefficient scaled by a scalingfactor. The scaling factor can include a ratio of the second aperturedimension to the first aperture dimension, where the ratio raised to thepower of a factor comprising a radial degree of the coefficient. In somecases, the aperture dimension includes an aperture radius. In somecases, the optical system include an optical tissue of a patient, andthe aperture dimension includes a pupil dimension. In some cases, thepupil dimension includes a pupil radius. Optionally, the first aperturedimension may be greater than the second aperture dimension.

In another aspect, embodiments of the present invention provide a methodof determining an optical surface model for an optical tissue system ofan eye. The method can include inputting a first optical datacorresponding to the optical tissue system of the eye, where the firstoptical data includes a first set of normalized Zernike expansioncoefficients corresponding to a first pupil radius of the eye. Themethod can also include calculating a second optical data correspondingto the optical tissue of the eye, where the second optical data includesa second set of normalized Zernike expansion coefficients correspondingto a second pupil radius of the eye. The second set of normalizedZernike expansion coefficient can be calculated based on the first setof normalized Zernike expansion coefficients scaled by a scaling factor.The scaling factor can include a ratio of the second aperture dimensionto the first aperture dimension. The ratio can be raised to the power ofa factor that includes a radial degree of the coefficient. The methodcan further include determining the optical surface model based on thesecond optical data. Optionally, the first pupil radius of the eye canbe greater than the second pupil radius of the eye.

In another aspect, embodiments of the present invention provide a systemfor calculating a modified normalized Zernike expansion coefficient foran optical system. The system can include, for example, means forinputting an original normalized Zernike expansion coefficient for theoptical system, where the original normalized Zernike expansioncoefficient is associated with a first aperture dimension. The systemcan also include means for calculating a modified normalized Zernikeexpansion coefficient for the optical system, where the modifiednormalized Zernike expansion coefficient is associated with a secondaperture dimension. The modified normalized Zernike expansioncoefficient can be calculated based on the original normalized Zernikeexpansion coefficient scaled by a scaling factor. The scaling factor caninclude a ratio of the second aperture dimension to the first aperturedimension, where the ratio raised to the power of a factor comprising aradial degree of the coefficient. The aperture dimension can include anaperture radius. In some cases, the optical system includes an opticaltissue of a patient, and the aperture dimension includes a pupildimension. Optionally, the pupil dimension may include a pupil radius.In some cases, the first aperture dimension is greater than the secondaperture dimension.

In another aspect, embodiments of the present invention provide a methodof calculating effective powers of an optical system. The method caninclude, for example, calculating a first effective power using a firstZernike expansion coefficient for the optical system, where the firstZernike expansion coefficient is associated with a first aperturedimension. The method can also include calculating a second effectivepower using a second Zernike expansion coefficient for the opticalsystem, where the second Zernike expansion coefficient is associatedwith a second aperture dimension. The second Zernike expansioncoefficient can be scaled relative to the first Zernike expansioncoefficient using a scaling factor that includes a ratio of the secondaperture dimension to the first aperture dimension raised to a power ofa factor that includes a radial degree of the coefficient. In somecases, the optical system includes an optical tissue of a patient, andthe aperture dimension includes a pupil dimension. In some cases, thepupil dimension includes a pupil radius.

In another aspect, embodiments of the present invention provide a systemfor calculating effective power for an optical system. The system caninclude means for inputting a first Zernike expansion coefficient forthe optical system, where the first Zernike expansion coefficient isassociated with a first aperture dimension. The system can also includemeans for calculating a second effective power using a second Zernikeexpansion coefficient for the optical system, where the second Zernikeexpansion coefficient is associated with a second aperture dimension andscaled relative to the first Zernike expansion coefficient using ascaling factor that includes a ratio of the second aperture dimension tothe first aperture dimension raised to a power of a factor that includesa radial degree of the coefficient. In some cases, the optical systemincludes an optical tissue of a patient, and the aperture dimensionincludes a pupil dimension. In some cases, the pupil dimension includesa pupil radius.

In a further aspect, embodiments of the present invention provide acomputer program product for determining an optical surface model for anoptical tissue system of an eye. The computer program product caninclude, for example, code for accepting a first optical datacorresponding to the optical tissue system of the eye, where the firstoptical data includes a first set of normalized Zernike expansioncoefficients corresponding to a first pupil radius of the eye. Theproduct can also include code for calculating a second optical datacorresponding to the optical tissue of the eye, where the second opticaldata includes a second set of normalized Zernike expansion coefficientscorresponding to a second pupil radius of the eye. According to code,the second set of normalized Zernike expansion coefficient can becalculated based on the first set of normalized Zernike expansioncoefficients scaled by a scaling factor. The scaling factor can includea ratio of the second aperture dimension to the first aperturedimension, and the ratio can be raised to the power of a factorcomprising a radial degree of the coefficient. the product can alsoinclude code for determining the optical surface model based on thesecond optical data. Optionally, the product includes acomputer-readable medium for storing the codes. In some cases, theoptical system includes an optical tissue of a patient, and the aperturedimension includes a pupil dimension.

For a fuller understanding of the nature and advantages of the presentinvention, reference should be had to the ensuing detailed descriptiontaken in conjunction with the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates a laser ablation system which may be included or usedwith an embodiment of the present invention.

FIG. 2 illustrates a simplified computer system according to anembodiment of the present invention.

FIG. 3 illustrates a wavefront measurement system according to anembodiment of the present invention.

FIG. 3A illustrates another wavefront measurement system according to anembodiment of the present invention.

FIG. 4 is a schematic block diagram illustrating software and/orhardware modules which may be included in the computer system of FIG. 2for use in embodiments of the invention.

FIG. 5 is a flowchart schematically illustrating an embodiment of amethod of the invention.

FIGS. 6A and 6B are contour plots of a wavefront at two different pupilsizes.

DETAILED DESCRIPTION OF THE INVENTION

The present invention can be readily adapted for use with existing lasersystems, wavefront measurement systems, and other optical measurementdevices. While the systems, software, and methods of the presentinvention are described primarily in the context of a laser eye surgerysystem, it should be understood the present invention may be adapted foruse in alternative eye treatment procedures and systems such asspectacle lenses, intraocular lenses, contact lenses, corneal ringimplants, collagenous corneal tissue thermal remodeling, and the like.

Turning now to the drawings, FIG. 1 illustrates a laser eye surgerysystem 10 of the present invention, including a laser 12 that produces alaser beam 14. Laser 12 is optically coupled to laser delivery optics16, which directs laser beam 14 to an eye E of patient P. A deliveryoptics support structure (not shown here for clarity) extends from aframe 18 supporting laser 12. A microscope 20 is mounted on the deliveryoptics support structure, the microscope often being used to image acornea of eye E.

Laser 12 generally comprises an excimer laser, ideally comprising anargon-fluorine laser producing pulses of laser light having a wavelengthof approximately 193 nm. Laser 12 will preferably be designed to providea feedback stabilized fluence at the patient's eye, delivered viadelivery optics 16. The present invention may also be useful withalternative sources of ultraviolet or infrared radiation, particularlythose adapted to controllably ablate the corneal tissue without causingsignificant damage to adjacent and/or underlying tissues of the eye.Such sources include, but are not limited to, solid state lasers andother devices which can generate energy in the ultraviolet wavelengthbetween about 185 and 205 nm and/or those which utilizefrequency-multiplying techniques. Hence, although an excimer laser isthe illustrative source of an ablating beam, other lasers may be used inthe present invention.

Laser system 10 will generally include a computer or programmableprocessor 22. Processor 22 may comprise (or interface with) aconventional PC system including the standard user interface devicessuch as a keyboard, a display monitor, and the like. Processor 22 willtypically include an input device such as a magnetic or optical diskdrive, an internet connection, or the like. Such input devices willoften be used to download a computer executable code from a tangiblestorage media 29 embodying any of the methods of the present invention.Tangible storage media 29 may take the form of a floppy disk, an opticaldisk, a data tape, a volatile or non-volatile memory, RAM, or the like,and the processor 22 will include the memory boards and other standardcomponents of modern computer systems for storing and executing thiscode. Tangible storage media 29 may optionally embody wavefront sensordata, wavefront gradients, a wavefront elevation map, a treatment map, acorneal elevation map, and/or an ablation table. While tangible storagemedia 29 will often be used directly in cooperation with a input deviceof processor 22, the storage media may also be remotely operativelycoupled with processor by means of network connections such as theinternet, and by wireless methods such as infrared, Bluetooth, or thelike.

Laser 12 and delivery optics 16 will generally direct laser beam 14 tothe eye of patient P under the direction of a computer 22. Computer 22will often selectively adjust laser beam 14 to expose portions of thecornea to the pulses of laser energy so as to effect a predeterminedsculpting of the cornea and alter the refractive characteristics of theeye. In many embodiments, both laser beam 14 and the laser deliveryoptical system 16 will be under computer control of processor 22 toeffect the desired laser sculpting process, with the processor effecting(and optionally modifying) the pattern of laser pulses. The pattern ofpulses may by summarized in machine readable data of tangible storagemedia 29 in the form of a treatment table, and the treatment table maybe adjusted according to feedback input into processor 22 from anautomated image analysis system in response to feedback data providedfrom an ablation monitoring system feedback system. Optionally, thefeedback may be manually entered into the processor by a systemoperator. Such feedback might be provided by integrating the wavefrontmeasurement system described below with the laser treatment system 10,and processor 22 may continue and/or terminate a sculpting treatment inresponse to the feedback, and may optionally also modify the plannedsculpting based at least in part on the feedback. Measurement systemsare further described in U.S. Pat. No. 6,315,413, the full disclosure ofwhich is incorporated herein by reference.

Laser beam 14 may be adjusted to produce the desired sculpting using avariety of alternative mechanisms. The laser beam 14 may be selectivelylimited using one or more variable apertures. An exemplary variableaperture system having a variable iris and a variable width slit isdescribed in U.S. Pat. No. 5,713,892, the full disclosure of which isincorporated herein by reference. The laser beam may also be tailored byvarying the size and offset of the laser spot from an axis of the eye,as described in U.S. Pat. Nos. 5,683,379, 6,203,539, and 6,331,177, thefull disclosures of which are incorporated herein by reference.

Still further alternatives are possible, including scanning of the laserbeam over the surface of the eye and controlling the number of pulsesand/or dwell time at each location, as described, for example, by U.S.Pat. No. 4,665,913, the full disclosure of which is incorporated hereinby reference; using masks in the optical path of laser beam 14 whichablate to vary the profile of the beam incident on the cornea, asdescribed in U.S. Pat. No. 5,807,379, the full disclosure of which isincorporated herein by reference; hybrid profile-scanning systems inwhich a variable size beam (typically controlled by a variable widthslit and/or variable diameter iris diaphragm) is scanned across thecornea; or the like. The computer programs and control methodology forthese laser pattern tailoring techniques are well described in thepatent literature.

Additional components and subsystems may be included with laser system10, as should be understood by those of skill in the art. For example,spatial and/or temporal integrators may be included to control thedistribution of energy within the laser beam, as described in U.S. Pat.No. 5,646,791, the full disclosure of which is incorporated herein byreference. Ablation effluent evacuators/filters, aspirators, and otherancillary components of the laser surgery system are known in the art.Further details of suitable systems for performing a laser ablationprocedure can be found in commonly assigned U.S. Pat. Nos. 4,665,913,4,669,466, 4,732,148, 4,770,172, 4,773,414, 5,207,668, 5,108,388,5,219,343, 5,646,791 and 5,163,934, the complete disclosures of whichare incorporated herein by reference. Suitable systems also includecommercially available refractive laser systems such as thosemanufactured and/or sold by Alcon, Bausch & Lomb, Nidek, WaveLight,LaserSight, Schwind, Zeiss-Meditec, and the like. Basis data can befurther characterized for particular lasers or operating conditions, bytaking into account localized environmental variables such astemperature, humidity, airflow, and aspiration.

FIG. 2 is a simplified block diagram of an exemplary computer system 22that may be used by the laser surgical system 10 of the presentinvention. Computer system 22 typically includes at least one processor52 which may communicate with a number of peripheral devices via a bussubsystem 54. These peripheral devices may include a storage subsystem56, comprising a memory subsystem 58 and a file storage subsystem 60,user interface input devices 62, user interface output devices 64, and anetwork interface subsystem 66. Network interface subsystem 66 providesan interface to outside networks 68 and/or other devices, such as thewavefront measurement system 30.

User interface input devices 62 may include a keyboard, pointing devicessuch as a mouse, trackball, touch pad, or graphics tablet, a scanner,foot pedals, a joystick, a touchscreen incorporated into the display,audio input devices such as voice recognition systems, microphones, andother types of input devices. User input devices 62 will often be usedto download a computer executable code from a tangible storage media 29embodying any of the methods of the present invention. In general, useof the term “input device” is intended to include a variety ofconventional and proprietary devices and ways to input information intocomputer system 22.

User interface output devices 64 may include a display subsystem, aprinter, a fax machine, or non-visual displays such as audio outputdevices. The display subsystem may be a cathode ray tube (CRT), aflat-panel device such as a liquid crystal display (LCD), a projectiondevice, or the like. The display subsystem may also provide a non-visualdisplay such as via audio output devices. In general, use of the term“output device” is intended to include a variety of conventional andproprietary devices and ways to output information from computer system22 to a user.

Storage subsystem 56 can store the basic programming and data constructsthat provide the functionality of the various embodiments of the presentinvention. For example, a database and modules implementing thefunctionality of the methods of the present invention, as describedherein, may be stored in storage subsystem 56. These software modulesare generally executed by processor 52. In a distributed environment,the software modules may be stored on a plurality of computer systemsand executed by processors of the plurality of computer systems. Storagesubsystem 56 typically comprises memory subsystem 58 and file storagesubsystem 60.

Memory subsystem 58 typically includes a number of memories including amain random access memory (RAM) 70 for storage of instructions and dataduring program execution and a read only memory (ROM) 72 in which fixedinstructions are stored. File storage subsystem 60 provides persistent(non-volatile) storage for program and data files, and may includetangible storage media 29 (FIG. 1) which may optionally embody wavefrontsensor data, wavefront gradients, a wavefront elevation map, a treatmentmap, and/or an ablation table. File storage subsystem 60 may include ahard disk drive, a floppy disk drive along with associated removablemedia, a Compact Digital Read Only Memory (CD-ROM) drive, an opticaldrive, DVD, CD-R, CD-RW, solid-state removable memory, and/or otherremovable media cartridges or disks. One or more of the drives may belocated at remote locations on other connected computers at other sitescoupled to computer system 22. The modules implementing thefunctionality of the present invention may be stored by file storagesubsystem 60.

Bus subsystem 54 provides a mechanism for letting the various componentsand subsystems of computer system 22 communicate with each other asintended. The various subsystems and components of computer system 22need not be at the same physical location but may be distributed atvarious locations within a distributed network. Although bus subsystem54 is shown schematically as a single bus, alternate embodiments of thebus subsystem may utilize multiple busses.

Computer system 22 itself can be of varying types including a personalcomputer, a portable computer, a workstation, a computer terminal, anetwork computer, a control system in a wavefront measurement system orlaser surgical system, a mainframe, or any other data processing system.Due to the ever-changing nature of computers and networks, thedescription of computer system 22 depicted in FIG. 2 is intended only asa specific example for purposes of illustrating one embodiment of thepresent invention. Many other configurations of computer system 22 arepossible having more or less components than the computer systemdepicted in FIG. 2.

Referring now to FIG. 3, one embodiment of a wavefront measurementsystem 30 is schematically illustrated in simplified form. In verygeneral terms, wavefront measurement system 30 is configured to senselocal slopes of a gradient map exiting the patient's eye. Devices basedon the Hartmann-Shack principle generally include a lenslet array tosample the gradient map uniformly over an aperture, which is typicallythe exit pupil of the eye. Thereafter, the local slopes of the gradientmap are analyzed so as to reconstruct the wavefront surface or map.

More specifically, one wavefront measurement system 30 includes an imagesource 32, such as a laser, which projects a source image throughoptical tissues 34 of eye E so as to form an image 44 upon a surface ofretina R. The image from retina R is transmitted by the optical systemof the eye (e.g., optical tissues 34) and imaged onto a wavefront sensor36 by system optics 37. The wavefront sensor 36 communicates signals toa computer system 22′ for measurement of the optical errors in theoptical tissues 34 and/or determination of an optical tissue ablationtreatment program. Computer 22′ may include the same or similar hardwareas the computer system 22 illustrated in FIGS. 1 and 2. Computer system22′ may be in communication with computer system 22 that directs thelaser surgery system 10, or some or all of the components of computersystem 22, 22′ of the wavefront measurement system 30 and laser surgerysystem 10 may be combined or separate. If desired, data from wavefrontsensor 36 may be transmitted to a laser computer system 22 via tangiblemedia 29, via an I/O port, via an networking connection 66 such as anintranet or the Internet, or the like.

Wavefront sensor 36 generally comprises a lenslet array 38 and an imagesensor 40. As the image from retina R is transmitted through opticaltissues 34 and imaged onto a surface of image sensor 40 and an image ofthe eye pupil P is similarly imaged onto a surface of lenslet array 38,the lenslet array separates the transmitted image into an array ofbeamlets 42, and (in combination with other optical components of thesystem) images the separated beamlets on the surface of sensor 40.Sensor 40 typically comprises a charged couple device or “CCD,” andsenses the characteristics of these individual beamlets, which can beused to determine the characteristics of an associated region of opticaltissues 34. In particular, where image 44 comprises a point or smallspot of light, a location of the transmitted spot as imaged by a beamletcan directly indicate a local gradient of the associated region ofoptical tissue.

Eye E generally defines an anterior orientation ANT and a posteriororientation POS. Image source 32 generally projects an image in aposterior orientation through optical tissues 34 onto retina R asindicated in FIG. 3. Optical tissues 34 again transmit image 44 from theretina anteriorly toward wavefront sensor 36. Image 44 actually formedon retina R may be distorted by any imperfections in the eye's opticalsystem when the image source is originally transmitted by opticaltissues 34. Optionally, image source projection optics 46 may beconfigured or adapted to decrease any distortion of image 44.

In some embodiments, image source optics 46 may decrease lower orderoptical errors by compensating for spherical and/or cylindrical errorsof optical tissues 34. Higher order optical errors of the opticaltissues may also be compensated through the use of an adaptive opticelement, such as a deformable mirror (described below). Use of an imagesource 32 selected to define a point or small spot at image 44 uponretina R may facilitate the analysis of the data provided by wavefrontsensor 36. Distortion of image 44 may be limited by transmitting asource image through a central region 48 of optical tissues 34 which issmaller than a pupil 50, as the central portion of the pupil may be lessprone to optical errors than the peripheral portion. Regardless of theparticular image source structure, it will be generally be beneficial tohave a well-defined and accurately formed image 44 on retina R.

In one embodiment, the wavefront data may be stored in a computerreadable medium 29 or a memory of the wavefront sensor system 30 in twoseparate arrays containing the x and y wavefront gradient valuesobtained from image spot analysis of the Hartmann-Shack sensor images,plus the x and y pupil center offsets from the nominal center of theHartmann-Shack lenslet array, as measured by the pupil camera 52 (FIG.3) image. Such information contains all the available information on thewavefront error of the eye and is sufficient to reconstruct thewavefront or any portion of it. In such embodiments, there is no need toreprocess the Hartmann-Shack image more than once, and the data spacerequired to store the gradient array is not large. For example, toaccommodate an image of a pupil with an 8 mm diameter, an array of a20×20 size (i.e., 400 elements) is often sufficient. As can beappreciated, in other embodiments, the wavefront data may be stored in amemory of the wavefront sensor system in a single array or multiplearrays.

While the methods of the present invention will generally be describedwith reference to sensing of an image 44, it should be understood that aseries of wavefront sensor data readings may be taken. For example, atime series of wavefront data readings may help to provide a moreaccurate overall determination of the ocular tissue aberrations. As theocular tissues can vary in shape over a brief period of time, aplurality of temporally separated wavefront sensor measurements canavoid relying on a single snapshot of the optical characteristics as thebasis for a refractive correcting procedure. Still further alternativesare also available, including taking wavefront sensor data of the eyewith the eye in differing configurations, positions, and/ororientations. For example, a patient will often help maintain alignmentof the eye with wavefront measurement system 30 by focusing on afixation target, as described in U.S. Pat. No. 6,004,313, the fulldisclosure of which is incorporated herein by reference. By varying aposition of the fixation target as described in that reference, opticalcharacteristics of the eye may be determined while the eye accommodatesor adapts to image a field of view at a varying distance and/or angles.

The location of the optical axis of the eye may be verified by referenceto the data provided from a pupil camera 52. In the exemplaryembodiment, a pupil camera 52 images pupil 50 so as to determine aposition of the pupil for registration of the wavefront sensor datarelative to the optical tissues.

An alternative embodiment of a wavefront measurement system isillustrated in FIG. 3A. The major components of the system of FIG. 3Aare similar to those of FIG. 3. Additionally, FIG. 3A includes anadaptive optical element 98 in the form of a deformable mirror. Thesource image is reflected from deformable mirror 98 during transmissionto retina R, and the deformable mirror is also along the optical pathused to form the transmitted image between retina R and imaging sensor40. Deformable mirror 98 can be controllably deformed by computer system22 to limit distortion of the image formed on the retina or ofsubsequent images formed of the images formed on the retina, and mayenhance the accuracy of the resultant wavefront data. The structure anduse of the system of FIG. 3A are more fully described in U.S. Pat. No.6,095,651, the full disclosure of which is incorporated herein byreference.

The components of an embodiment of a wavefront measurement system formeasuring the eye and ablations may comprise elements of a VISXWaveScan® system, available from VISX, INCORPORATED of Santa Clara,Calif. One embodiment includes a WaveScan® with a deformable mirror asdescribed above. An alternate embodiment of a wavefront measuring systemis described in U.S. Pat. No. 6,271,915, the full disclosure of which isincorporated herein by reference. It is appreciated that any wavefrontaberrometer could be employed for use with the present invention.

FIGS. 4 and 5 schematically illustrate embodiments of hardware and/orsoftware modules of computer system 22 and a related method,respectively. These embodiments can generate scaled wavefrontreconstruction data suitable for analysis of a patient's eye when apupil of the patient changes size from a relatively large wavefrontmeasurement pupil size to a smaller size. Structures and methods forreconstructing a wavefront and/or generating prescriptions fromwavefront data are well documented in a variety of patent literature,including U.S. patent application Ser. No. 10/738,358, as filed on Dec.5, 2003 and entitled “Presbyopia Correction Using Patient Data;” andSer. No. 11/134,630, as filed on May 19, 2005 and entitled “ResidualAccommodation Threshold for Correction of Presbyopia and OtherPresbyopia Correction Using Patient Data,” the full disclosures of whichare incorporated herein by reference. The following description willaddress scaling of the wavefront data, particularly scaling of Zernikepolynomial expansion coefficients of a wavefront so as to accommodate ormodel constriction of the pupil of the patient when viewing objectsunder different lighting conditions, differing viewing distances, andthe like.

If W(Rr, θ) represents the ocular aberrations of a human eye measured asthe optical path difference, the wavefront can be decomposed into a setof complete and orthogonal basis functions as:

$\begin{matrix}{{{W\left( {{Rr},\theta} \right)} = {\sum\limits_{i = 0}^{\infty}{a_{i}{F_{i}\left( {r,\theta} \right)}}}},} & \left( {{Eq}.\mspace{14mu} 1} \right)\end{matrix}$

where a_(i) is the coefficient of the ith basis function F_(i)(r, θ) andR is the pupil radius. Here, r is the radial variable in the polarcoordinates defining the unit circle. Zernike polynomials have beenwidely used as a set of basis functions because of their connection toclassical aberrations in optical systems with circular apertures.

The normalized Zernike polynomials may be defined as:Z _(i)(r,θ)=R _(n) ^({|m|})(r)θ^(m)(θ),  (Eq. 2)where n and m denote the radial degree and the azimuthal frequency,respectively; the radial polynomials are defined as:

$\begin{matrix}{{R_{n}^{\{{m}\}}(r)} = {\sum\limits_{s = 0}^{{({n - {m}})}/2}{\frac{\left( {- 1} \right)^{s}\sqrt{n + 1}{\left( {n - s} \right)!}}{{{{{s!}\left\lbrack {{\left( {n + m} \right)/2} - s} \right\rbrack}!}\left\lbrack {{\left( {n - m} \right)/2} - s} \right\rbrack}!}r^{n - {2s}}}}} & \left( {{Eq}.\mspace{14mu} 3} \right)\end{matrix}$and the triangular functions as:

$\begin{matrix}{{\theta^{m}(\theta)} = \left\{ \begin{matrix}{\sqrt{2}\cos{m}\theta} & \left( {m > 0} \right) \\1 & \left( {m = 0} \right) \\{\sqrt{2}\sin{m}\theta} & \left( {m < 0} \right)\end{matrix} \right.} & \left( {{Eq}.\mspace{14mu} 4} \right)\end{matrix}$

Both the single-index i and the double-index m and n may be referred toherein. These two different indexing schemes can be effectivelyidentical. For example, the decision whether to use a single or doubleindex may be based on convenience.

It can be usefully assumed that (1) the optical properties of the humaneye do not change when the pupil constricts and (2) the constriction ofthe pupil is concentric.

Assume that the pupil aperture changes from R₁ to R₂, as shown in FIGS.6A and 6B. The ocular aberrations of the pupil with the smaller radius,R₂, are the same as the aberrations of the area defined by radius R₂when the pupil size is R₁; i.e., the aberrations do not change when thepupil size changes.

FIGS. 6A and 6B graphically illustrate contour plots of a wavefront mapwith pupil radius R₁ (in FIG. 6A) and the wavefront map when the pupilsize constricts to pupil radius R₁ to R₂ (in FIG. 6B). The two maps arein the same scale. Units are in micrometers of optical path difference.Note that the portion of the wavefront defined by R₂ in FIG. 6A is thesame as the plot in FIG. 6B.

If F_(i)(r, θ) is replaced by Zernike polynomials Z_(i)(r, θ) in Eq.(1), the entire wavefront over the pupil with radius R₁ can be writtenas:

$\begin{matrix}{{{W_{1}\left( {{R_{1}r},\theta} \right)} = {\sum\limits_{i = 0}^{\infty}{a_{i}{Z_{i}\left( {r,\theta} \right)}}}},} & \left( {{Eq}.\mspace{14mu} 5} \right)\end{matrix}$where a_(i) is the ith Zernike coefficient representing the Zernikeexpansion into the pupil when the pupil radius is R₁. Similarly, theentire wavefront over the pupil with radius R₂ can be written as:

$\begin{matrix}{{{W_{2}\left( {{R_{2}r},\theta} \right)} = {\sum\limits_{i = 0}^{\infty}{b_{i}{Z_{i}\left( {{\in r},\theta} \right)}}}},} & \left( {{Eq}.\mspace{14mu} 6} \right)\end{matrix}$where b_(i) is the ith Zernike coefficient representing the Zernikeexpansion into the pupil when the pupil radius is R₂. The next step isto determine the relationship of {b_(i)} to {a_(i)}.

From Eq. (5), to represent only the area defined by radius R₂,W′₁(R_(i)r,θ), only r (which runs from 0 to 1) needs to be scaled by ascaling factor of ε r, which runs from 0 to ε. Therefore,

$\begin{matrix}{{W_{1}^{\prime}\left( {{R_{1}r},\theta} \right)} = {\sum\limits_{i = 0}^{\infty}{a_{i}{{Z_{i}\left( {{\in r},\theta} \right)}.}}}} & \left( {{Eq}.\mspace{14mu} 7} \right)\end{matrix}$As shown in FIGS. 6A and B, it is known thatW ₁′(R ₁ r,θ)=W ₂(R ₂ r,θ)  (Eq. 8)And so, from Eqs. (6)-(8), it is found that

$\begin{matrix}{{\sum\limits_{i = 0}^{\infty}{b_{i}{Z_{i}\left( {r,\theta} \right)}}} = {\sum\limits_{i = 0}^{\infty}{a_{i}{{Z_{i}\left( {{\in r},\theta} \right)}.}}}} & \left( {{Eq}.\mspace{14mu} 9} \right)\end{matrix}$

Derivation of Eq. (9) comes from the definition of wavefront expansioninto basis functions. Equation (9) can be applied to any set of basisfunctions. If the triangular function is the same in both sides of Eq.(9), i.e., there is no rotation, after Eq. (2) is applied therelationship between the sets of coefficients {a_(i)} and {b_(i)} is

$\begin{matrix}{{\sum\limits_{n}{\sum\limits_{m}{b_{n}^{m}{R_{n}^{\{{m}\}}(r)}}}} = {\sum\limits_{n}{\sum\limits_{m}{a_{n}^{m}{{R_{n}^{\{{m}\}}\left( {\in r} \right)}.}}}}} & \left( {{Eq}.\mspace{14mu} 10} \right)\end{matrix}$Substituting R_(n) ^(m)(r) from Eq. (3) to Eq. (10) yields

$\begin{matrix}{{{\sum\limits_{n = 0}^{N}{\sum\limits_{m}{b_{n}^{m}{\sum\limits_{s = 0}^{{({n - {m}})}/2}\frac{\left( {- 1} \right)^{s}\sqrt{n + 1}{\left( {n - s} \right)!}r^{n - {2s}}}{{{{{s!}\left\lbrack {{\left( {n + m} \right)/2} - s} \right\rbrack}!}\left\lbrack {{\left( {n - m} \right)/2} - s} \right\rbrack}!}}}}} = {\sum\limits_{n = 0}^{N}{\sum\limits_{m}{a_{n}^{m}{\sum\limits_{s = 0}^{{({n - {m}})}/2}\frac{{\left( {- 1} \right)^{s}\sqrt{n + 1}{\left( {n - s} \right)!}} \in^{n - {2s}}r^{n - {2s}}}{{{{{s!}\left\lbrack {{\left( {n + m} \right)/2} - s} \right\rbrack}!}\left\lbrack {{\left( {n - m} \right)/2} - s} \right\rbrack}!}}}}}},} & \left( {{Eq}.\mspace{14mu} 11} \right)\end{matrix}$where N is the total number of orders used for the expansion. Expandingthis equation into a radial series yields radial powers of r from 0 toN, resulting in (N+1) equations. This leads to a solution for (N+1)relations between a_(n) ^(m) and b_(n) ^(m).

Going from top to bottom for radial powers of r in Eq. (11), considerthe r^(N) case. We get r^(N) terms only when n=N and s=0. Then Eq. (11)results inb_(N) ^(m)=ε^(N)a_(N) ^(m).  (Eq. 12)

Similarly, for the r^(N-1) case, we get the terms only when n=N−1 ands=0. Then Eq. (11) yieldsb _(N-1) ^(m)=ε^(N-1) a _(N-1) ^(m)  (Eq. 13)

To obtain a general solution, consider the order n with azimuthalfrequency m. So far r^(n), we know that s=0 for order n, s=1 for ordern+2, s=2 for order n+4, . . . , or s=(N−n)/2 for order n+2[(N−n)/2] canhave the radial order of r^(n). Hence,

$\begin{matrix}{{{\sum\limits_{i = 0}^{{({N - n})}/2}{b_{n - {2i}}^{m}\frac{\left( {- 1} \right)^{i}\sqrt{n + {2i} + 1}{\left( {n + i} \right)!}r^{n}}{{{{{i!}\left\lbrack {{\left( {n + {2i} + m} \right)/2} - i} \right\rbrack}!}\left\lbrack {{\left( {n + {2i} - m} \right)/2} - i} \right\rbrack}!}}} = {\sum\limits_{i = 0}^{{({N - n})}/2}{a_{n + {2i}}^{m}\frac{{\left( {- 1} \right)^{i}\sqrt{n + {2i} + 1}{\left( {n - i} \right)!}} \in^{n}r^{n}}{{{{{i!}\left\lbrack {{\left( {n + {2i} + m} \right)/2} - i} \right\rbrack}!}\left\lbrack {{\left( {n + {2i} - m} \right)/2} - i} \right\rbrack}!}}}},} & \left( {{Eq}.\mspace{14mu} 14} \right)\end{matrix}$

In Eq. (14), index i was used to prevent confusion with index s used inEq. (11), although both i and s have the same summation structure.Because Eq. (14) was derived for r^(n) only, m can be any integer from−n to n with a step of 2.

Because the denominators at both sides of Eq. (14) are the same for anygiven i, then

$\begin{matrix}{{\sum\limits_{i = 0}^{{({N - n})}/2}{\frac{b_{n + {2i}}^{m}}{i!}\left( {- i} \right)^{i}\sqrt{n + {2i} + 1}\left( {n + i} \right)}}!={\sum\limits_{i = 0}^{{({N - n})}/2}{\frac{\in^{n}a_{n + {2i}}^{m}}{i!}\left( {- 1} \right)^{i}\sqrt{n + {2i} + 1}{\left( {n + i} \right)!}}}} & \left( {{Eq}.\mspace{14mu} 15} \right)\end{matrix}$

Expanding i=0 case from Eq. (15), we can obtain a recursive formula as

$\begin{matrix}{b_{n}^{m} = {{\in^{n}{a_{n}^{m} + {\sum\limits_{i = 1}^{{({N - n})}/2}{\frac{\left( {- 1} \right)^{i}}{{i!}{n!}}\sqrt{\frac{n + {2i} + 1}{n + 1}}{\left( {n + i} \right)!}\left( {\in^{n}{a_{n + {2i}}^{m} - b_{n + {2i}}^{m}}} \right)}}}} = {{\in^{n}{a_{n}^{m} + {\sum\limits_{i = 1}^{{({N - n})}/2}{\frac{\left( {- 1} \right)^{i}}{i!}\sqrt{\left( {n + {2i} + 1} \right)\left( {n + 1} \right)} \times \left( {n + i} \right)\left( {n + i - 1} \right)\mspace{14mu}\ldots\mspace{14mu}\left( {n + 2} \right)\left( {\in^{n}{a_{n + {2i}}^{m} - b_{n + {2i}}^{m}}} \right)}}}} = {\in^{n}{a_{n}^{m} + {\sum\limits_{i = 1}^{{({N - n})}/2}{\frac{\left( {- 1} \right)^{i}{\left( {n + i} \right)!}}{{\left( {n + 1} \right)!}{i!}}\sqrt{\left( {n + {2i} + 1} \right)\left( {n + 1} \right)} \times {\left( {\in^{n}{a_{n + {2i}}^{m} - b_{n + {2i}}^{m}}} \right).}}}}}}}} & \left( {{Eq}.\mspace{14mu} 16} \right)\end{matrix}$

Equation (16) is the final recursive formula. With the use of Eq. (16)and Eqs. (12) and (13), relations between {b_(i)} and {a_(i)} can beobtained analytically.

To obtain a nonrecursive formula, Eq. (16) is applied to replaceb_(n+2i) ^(m). For example, the coefficient of the (n+2)th order can bewritten as

$\begin{matrix}{b_{n + 2}^{m} = {\in^{n + 2}{a_{n + 2}^{m} + {\sum\limits_{i = 1}^{{{({N - n})}/2} - 1}{\frac{\left( {- 1} \right)^{i}{\left( {n + i + 2} \right)!}}{{\left( {n + 3} \right)!}{i!}} \times \sqrt{\left( {n + {2i} + 3} \right)\left( {n + 3} \right)}{\left( {\in^{n + 2}{a_{n + {2i} + 2}^{m} - b_{n + {2i} + 2}^{m}}} \right).}}}}}} & \left( {{Eq}.\mspace{14mu} 17} \right)\end{matrix}$

With the expansion of b_(n+2i) ^(m) to each order higher than n forb_(n+2i) in Eq. (16) and some laborious arithmetic (see the derivationof Eq. 18 below), a final analytical formula is obtained as

$\begin{matrix}{b_{n}^{m} = {\in^{n}{\left\lbrack {{a_{n}^{m} + {\sum\limits_{i = 1}^{{({N - n})}/2}{a_{n + {2i}}^{m}\sqrt{\left. {n + {2i} + 1} \right)\left( {n + 1} \right)} \times {\sum\limits_{j = 0}^{i}\frac{\left( {- 1} \right)^{i + j}{\left( {n + i + j} \right)!}}{{\left( {n + j + 1} \right)!}{\left( {i - j} \right)!}{j!}}}}}} \in^{2j}} \right\rbrack.}}} & \left( {{Eq}.\mspace{14mu} 18} \right)\end{matrix}$

Sometimes it is useful to use unnormalized Zernike polynomials. In thiscase, Eq. (18) can be derived (see the derivation of Eq. 19 below) as

$\begin{matrix}{b_{n}^{m} = {\in^{n}{\quad{\left\lbrack {{a_{n}^{m} + {\left( {n + 1} \right){\sum\limits_{i = 1}^{{({N - n})}/2}{a_{n + {2i}}^{m}{\sum\limits_{j = 0}^{i}\frac{\left( {- 1} \right)^{i + j}{\left( {n + i + j} \right)!}}{{\left( {n + j + 1} \right)!}{\left( {i - j} \right)!}{j!}}}}}}} \in^{2j}} \right\rbrack.}}}} & \left( {{Eq}.\mspace{14mu} 19} \right)\end{matrix}$

The dioptric power of human eyes is typically the power desired from athin lens with a uniform optical power to give the subject the bestdistance vision. This dioptric power may be independent of pupil size.However, if high-order aberrations are present, especially radiallysymmetrical terms, dioptric power can change when pupil size changes.The instantaneous power that is dependent on pupil size is herein calledeffective power.

Because of the aberration-balancing nature of Zernike polynomials, allsymmetrical terms are balanced to give a minimum root-means-squareerror. And so, an aspherical optical surface represented by differenthigh-order spherical aberrations can be used to increase the depth offield. Therefore, the effective power can be defined only by the defocusterm as

$\begin{matrix}{{P_{eff} = {- \frac{4\sqrt{3}a_{2}^{0}}{R^{2}}}},} & \left( {{Eq}.\mspace{14mu} 20} \right)\end{matrix}$where R stands for the instantaneous pupil radius in millimeters whenthe instantaneous coefficient of defocus term a₂ ⁰ is given inmicrometers to get the effective power of the diopters. If a wavefrontmap is defined in radius R with a set of Zernike polynomials, when thepupil constricts, the smaller map is then redefined with a new set ofZernike polynomials, and it will have a set of Zernike coefficients thatis different from the original set.

If only the defocus term (n=2, m=0) is considered, Eq. (18) becomes

$\begin{matrix}{b_{2}^{0} = {\in^{2}{\left\lbrack {{a_{2}^{0}{\sum\limits_{i = 1}^{{N/2} - 1}{a_{2{({i + 1})}}^{0}\sqrt{3\left( {3 + {2i}} \right)}{\sum\limits_{j = 0}^{i}\frac{\left( {- 1} \right)^{i + j}{\left( {i + j + 2} \right)!}}{{\left( {i - j} \right)!}{\left( {j + 3} \right)!}{j!}}}}}} \in^{2j}} \right\rbrack.}}} & \left( {{Eq}.\mspace{14mu} 21} \right)\end{matrix}$

Taking a₂ ⁰ as b₂ ⁰ and R as ε R in Eq. (20) and using Eq. (21), theeffective power becomes

$\begin{matrix}{{P_{eff}( \in )} = {- {{\frac{4\sqrt{3}}{R^{2}}\left\lbrack {{a_{2}^{0} + {\sum\limits_{i = 1}^{{N/2} - 1}{a_{2{({i + 1})}}^{0}\sqrt{3\left( {3 + {2i}} \right)} \times {\sum\limits_{j = 0}^{i}\frac{\left( {- 1} \right)^{i + j}{\left( {i + j + 2} \right)!}}{{\left( {i - j} \right)!}{\left( {j + 3} \right)!}{j!}}}}}} \in^{2j}} \right\rbrack}.}}} & \left( {{Eq}.\mspace{14mu} 22} \right)\end{matrix}$

Sometimes wavefronts of radially symmetric aspheric optical surfaces arenot represented by Zernike polynomials but by power series. In thiscase, the wavefront is written as

$\begin{matrix}{{{W({Rr})} = {\sum\limits_{n = 0}^{N/2}{a_{2n}r^{2n}}}},} & \left( {{Eq}.\mspace{14mu} 23} \right)\end{matrix}$where R stands for the pupil radius, r is the radial variable in polarcoordinates that defines the unit circle, and N is the maximum radialpower. The coefficients {a_(2n)} of the power series of Eq. (23) can beconverted into Zernike polynomials so that the effective power can bewritten as

$\begin{matrix}{{P_{eff}( \in )} = {{- \frac{12}{R^{2}}}{\sum\limits_{n = 1}^{N/2}{\frac{n \in^{2{({n - 1})}}}{\left( {n + 1} \right)\left( {n + 2} \right)}{a_{2n}.}}}}} & \left( {{Eq}.\mspace{14mu} 24} \right)\end{matrix}$

The influence of spherical aberration on refraction may not havepreviously been quantified analytically. Equation (22) indicates thatwhen higher-order spherical aberrations exist, the effective power is nolonger determined only by the defocus term. With Eqs. (22) and (24), itis now possible to evaluate the influence of an aspheric shape onrefraction.

A more intuitive approach than that described previously was above usedto derive a simpler and nonrecursive formula for calculating a new setof Zernike polynomial expansion coefficients for ocular aberrations whenthe pupil constricts. A relationship has been established between theeffective power and the high-order radially symmetrical terms that canbe useful for determining the influence of high-order sphericalaberrations on refraction. Some or all of the approaches described abovemay be embodied in methods and systems for measuring opticalcharacteristics (often including wavefront data) of eyes and otheroptical systems, for generating desired refractive changes (includingprescriptions), and/or for implementing refractive changes (includinglaser eye surgery, contact lenses, intraocular lenses, and the like).Such embodiments may optionally include (and/or make use of) some or allof the structures described above regarding FIGS. 1-3A, optionally perthe exemplary embodiments of FIGS. 4 and 5. A wide variety ofalternative embodiments may also be implemented, optionally using any ofthe wide variety of known eye measurement and refraction alteringtechniques, new eye measurement and refraction altering techniques whichare developed, or a combination of both. Exemplary embodiments may, forexample, be used for calculation of effective powers of an eye atdiffering pupil sizes and/or locations, including those induced bydiffering viewing distances (and/or other viewing conditions). Asexplained in more detail US Patent Publication No. 20040169820, the fulldisclosure of which is incorporated herein by reference, suchcalculations of effective powers may have advantages for treatment ofpresbyopia.

Derivation of Equation (18)

Begin with the formula for orders n+2. Expand Eq. (17) as

$\begin{matrix}{b_{n + 2}^{m} \in^{n + 2}{a_{n + 2}^{m} - {\sqrt{\left( {n + 5} \right)\left( {n + 3} \right)}\left( {\in^{n + 2}{a_{n + 4}^{m} - b_{n + 4}^{m}}} \right)} + {\frac{1}{2}\left( {n + 4} \right)\sqrt{\left( {n + 7} \right)\left( {n + 3} \right)}\left( {\in^{n + 2}{a_{n + 6}^{m} - b_{n + 6}^{m}}} \right)} - {\frac{1}{6}\left( {n + 5} \right) \times \left( {n + 4} \right)\sqrt{\left( {n + 9} \right)\left( {n + 3} \right)}\left( {\in^{n + 2}{a_{n + 8}^{m} - b_{n + 8}^{m}}} \right)} + \ldots + {\left( {- 1} \right)^{{{({N - n})}/2} - 1}{\frac{\sqrt{\left( {N + 1} \right)\left( {n + 3} \right)}}{\left\lbrack {{\left( {N - n} \right)/2} - 1} \right\rbrack!}\left\lbrack {{\left( {N + n} \right)/2} + 1} \right\rbrack} \times {\quad{{\left\lbrack {\left( {N + n} \right)/2} \right\rbrack{\ldots\left( {n + 5} \right)}\left( {n + 4} \right)\left( {\in^{n + 2}{a_{N}^{m} - b_{N}^{m}}} \right)},}}}}} & \left( {{{Eq}.\mspace{14mu} A}\; 1} \right)\end{matrix}$where it is assume that N−n is even. If N−n is odd, Eq. (A1) becomes

$\begin{matrix}{b_{n + 2}^{m} = {\in^{n + 2}{a_{n + 2}^{m} - {\sqrt{\left( {n + 5} \right)\left( {n + 3} \right)}\left( {\in^{n + 2}{a_{n + 4}^{m} - b_{n + 4}^{m}}} \right)} + {\frac{1}{2}\left( {n + 4} \right) \times \sqrt{\left( {n + 7} \right)\left( {n + 3} \right)}\left( {\in^{n + 2}{a_{n + 6}^{m} - b_{n + 6}^{m}}} \right)} - {\frac{1}{6}\left( {n + 5} \right)\left( {n + 4} \right) \times \sqrt{\left( {n + 9} \right)\left( {n + 3} \right)}\left( {\in^{n + 2}{a_{n + 8}^{m} - b_{n + 8}^{m}}} \right)} + \ldots + {\left( {- 1} \right)^{{({N - n - 3})}/2} \times {\frac{\sqrt{N\left( {n + 3} \right)}}{\left\lbrack {\left( {N - n - 3} \right)/2} \right\rbrack!}\left\lbrack {\left( {N + n + 1} \right)/2} \right\rbrack}{\quad{\left\lbrack {\left( {N + n - 1} \right)/2} \right\rbrack\ldots \times \left( {n + 5} \right)\left( {n + 4} \right)\left( {\in^{n + 2}{a_{N - 1}^{m} - b_{N - 1}^{m}}} \right)}}}}}} & \left( {{{Eq}.\mspace{14mu} A}\; 2} \right)\end{matrix}$

It is assumed for the next two formulas that N−n is even. If N−n is odd,an adjustment similar to that in Eq. (A2) can be done. To simplify theprocess, formulas for N−n being odd will not be given for the next twocases.

Similarly, for order n+4, the expression becomes

$\begin{matrix}{b_{n + 4}^{m} = {\in^{n + 4}{a_{n + 4}^{m} - {\sqrt{\left( {n + 7} \right)\left( {n + 5} \right)}\left( {\in^{n + 4}{a_{n + 6}^{m} - b_{n + 6}^{m}}} \right)} + {\frac{1}{2}\left( {n + 6} \right) \times \sqrt{\left( {n + 9} \right)\left( {n + 5} \right)}\left( {\in^{n + 4}{a_{n + 8}^{m} - b_{n + 8}^{m}}} \right)} - {\frac{1}{6}\left( {n + 7} \right)\left( {n + 6} \right) \times \sqrt{\left( {n + 11} \right)\left( {n + 5} \right)}\left( {\in^{n + 4}{a_{n + 10}^{m} - b_{n + 10}^{m}}} \right)} + \ldots + {\left( {- 1} \right)^{{{({N - n})}/2} - 2} \times {\frac{\sqrt{\left( {N + 1} \right)\left( {n + 5} \right)}}{\left\lbrack {{\left( {N - n} \right)/2} - 2} \right\rbrack!}\left\lbrack {{\left( {N + n} \right)/2} + 2} \right\rbrack}{\quad{\left\lbrack {{\left( {N + n} \right)/2} + 1} \right\rbrack\ldots \times \left( {n + 7} \right)\left( {n + 6} \right){\left( {\in^{n + 4}{a_{N}^{m} - b_{N}^{m}}} \right).}}}}}}} & \left( {{{Eq}.\mspace{14mu} A}\; 3} \right)\end{matrix}$

And for order n+6, the expression is

$\begin{matrix}{b_{n + 6}^{m} = {\in^{n + 6}{a_{n + 6}^{m} - {\sqrt{\left( {n + 9} \right)\left( {n + 7} \right)}\left( {\in^{n + 6}{a_{n + 8}^{m} - b_{n + 8}^{m}}} \right)} + {\frac{1}{2}\left( {n + 8} \right) \times \sqrt{\left( {n + 11} \right)\left( {n + 7} \right)}\left( {\in^{n + 6}{a_{n + 10}^{m} - b_{n + 10}^{m}}} \right)} - {\frac{1}{6}\left( {n + 9} \right)\left( {n + 8} \right) \times \sqrt{\left( {n + 11} \right)\left( {n + 7} \right)}\left( {\in^{n + 6}{a_{n + 12}^{m} - b_{n + 12}^{m}}} \right)} + \ldots + {\left( {- 1} \right)^{{{({N - n})}/2} - 3} \times {\frac{\sqrt{\left( {N + 1} \right)\left( {n + 7} \right)}}{\left\lbrack {{\left( {N - n} \right)/2} - 3} \right\rbrack!}\left\lbrack {{\left( {N + n} \right)/2} + 3} \right\rbrack}{\quad{\left\lbrack {{\left( {N + n} \right)/2} + 2} \right\rbrack\ldots \times \left( {n + 9} \right)\left( {n + 8} \right){\left( {\in^{n + 6}{a_{N}^{m} - b_{N}^{m}}} \right).}}}}}}} & \left( {{{Eq}.\mspace{14mu} A}\; 4} \right)\end{matrix}$

If this process continues, we would finally obtain either Eq. (12) orEq. (13) depending on whether N−n is even or odd. With the use of Eqs.(A1)-(A4), Eqs. (12) and (13), and combinations of terms for a_(n) ^(m),a_(n+2) ^(m), a_(n+4) ^(m), . . . , Eq. (16) becomes

$\begin{matrix}{b_{n}^{m} = {\in^{n}{a_{n}^{m} - \sqrt{\left( {n + 3} \right)\left( {n + 1} \right)}} \in^{n}{{\left( {{1 -} \in^{2}} \right)a_{n + 2}^{m}} + {{\frac{1}{2}\left\lbrack {{\left( {n + 2} \right) - {2\left( {n + 3} \right)}} \in^{2}{+ \left( {n + 4} \right)} \in^{4}} \right\rbrack} \times \sqrt{\left( {n + 5} \right)\left( {n + 1} \right)}}} \in^{n}{a_{n + 4}^{m} - {{\frac{1}{6}\left\lbrack {{{\left( {n + 2} \right)\left( {n + 3} \right)} - {3\left( {n + 3} \right)\left( {n + 4} \right)}} \in^{2}{{+ 3}\left( {n + 4} \right)\left( {n + 5} \right)} \in^{4}{{- \left( {n + 5} \right)}\left( {n + 6} \right)} \in^{6}} \right\rbrack}\sqrt{\left( {n + 7} \right)\left( {n + 1} \right)}}} \in^{n}{a_{n + 6}^{m} + {\ldots.}}}} & \left( {{{Eq}.\mspace{14mu} A}\; 5} \right)\end{matrix}$

Noticing that the fraction in each summation term of a Zernikecoefficient can be expressed as (−1)^(j)/j!, where j is the order of thesummation term, and that the number of expansion into ε with eachsummation is similar to a binomial expansion with an increasing numberof multiplication factors relating to n, we can express a finalanalytical formula as

$\begin{matrix}{b_{n}^{m} = {\in^{n}\left\lbrack {{a_{n}^{m}{\sum\limits_{i = 1}^{{({N - n})}/2}{a_{n + {2i}}^{m}\sqrt{\left( {n + {2i} + 1} \right)\left( {n + 1} \right)} \times {\sum\limits_{j = 0}^{i}\frac{\left( {- 1} \right)^{i + j}{\left( {n + i + j} \right)!}}{{\left( {n + j + 1} \right)!}{\left( {i - j} \right)!}{j!}}}}}} \in^{2j}} \right\rbrack}} & \left( {{{Eq}.\mspace{14mu} A}\; 6} \right)\end{matrix}$

Derivation of Equation (19)

The unnormalized Zernike polynomials can be written asZ _(i)(r,θ)=R _(n) ^({|m|})(r)θ^(m)(θ),  (Eq. B1)where the unnormalized Zernike radial polynomials are defined as

$\begin{matrix}{{R_{n}^{\{{m}\}}(r)} = {\sum\limits_{s = 0}^{{({n - {m}})}/2}{\frac{\left( {- 1} \right)^{s}{\left( {n - s} \right)!}}{{{{{s!}\left\lbrack {{\left( {n + m} \right)/2} - s} \right\rbrack}!}\left\lbrack {{\left( {n - m} \right)/2} - s} \right\rbrack}!}{r^{n - {2s}}.}}}} & \left( {{Eq}.\mspace{14mu}{B2}} \right)\end{matrix}$

Following a process similar to that described previously, the recursiveformula for unnormalized Zernike coefficients can be derived as

$\begin{matrix}{b_{n}^{m} = {\in^{n}{a_{n}^{m} + {\sum\limits_{i = 1}^{{({N - n})}/2}{\frac{\left( {- 1} \right)^{i}{\left( {n + i} \right)!}}{{n!}{i!}}{\left( {\in^{n}{a_{n + {2i}}^{m} - b_{n + {2i}}^{m}}} \right).}}}}}} & \left( {{Eq}.\mspace{14mu}{B3}} \right)\end{matrix}$

With the same process as described in Appendix A, a final nonrecursiveformula for unnormalized Zernike coefficients can be written as

$\begin{matrix}{b_{n}^{m} = {\in^{n}{\quad{\left\lbrack {{a_{n}^{m} + {\left( {n + 1} \right){\sum\limits_{i = 1}^{{({N - n})}/2}{a_{n + {2i}}^{m} \times {\sum\limits_{j = 0}^{i}\frac{\left( {- 1} \right)^{i + j}{\left( {n + i + j} \right)!}}{{\left( {n + j + 1} \right)!}{\left( {i - j} \right)!}{j!}}}}}}} \in^{2j}} \right\rbrack.}}}} & \left( {{Eq}.\mspace{14mu}{B4}} \right)\end{matrix}$

Each of the above calculations may be performed using a computer orother processor having hardware, software, and/or firmware. The variousmethod steps may be performed by modules, and the modules may compriseany of a wide variety of digital and/or analog data processing hardwareand/or software arranged to perform the method steps described herein.The modules optionally comprising data processing hardware adapted toperform one or more of these steps by having appropriate machineprogramming code associated therewith, the modules for two or more steps(or portions of two or more steps) being integrated into a singleprocessor board or separated into different processor boards in any of awide variety of integrated and/or distributed processing architectures.These methods and systems will often employ a tangible media embodyingmachine-readable code with instructions for performing the method stepsdescribed above. Suitable tangible media may comprise a memory(including a volatile memory and/or a non-volatile memory), a storagemedia (such as a magnetic recording on a floppy disk, a hard disk, atape, or the like; on an optical memory such as a CD, a CD-R/W, aCD-ROM, a DVD, or the like; or any other digital or analog storagemedia), or the like.

As noted above, a variety of output data can be generated by the systemsand methods of the present invention. Such outputs may be used for avariety of research, comparison, prediction, diagnostic, andverification operations. The outputs may be evaluated directly, or theymay be used as input into the system for further analysis. In someembodiments, the outputs will be used to model the effect of an oculartreatment prior to application. In other embodiments, the outputs willbe used to evaluate the effect of an ocular treatment after application.The outputs may also be used to design ocular treatments. Relatedly, itis possible to create treatment tables based on outputs of the instantinvention.

While exemplary embodiments have described in some detail for clarity ofunderstanding and by way of example, a variety of adaptations,modifications, and changes will be obvious of those of skill in the art.Hence, the scope of the present invention is limited solely by theappended claims.

1. A method of calculating a modified normalized Zernike expansioncoefficient for an optical system, the method comprising: inputting anoriginal normalized Zernike expansion coefficient for the opticalsystem, the original normalized Zernike expansion coefficient associatedwith a first aperture dimension; and calculating a modified normalizedZernike expansion coefficient for the optical system, the modifiednormalized Zernike expansion coefficient associated with a secondaperture dimension, wherein the modified normalized Zernike expansioncoefficient is calculated based on the original normalized Zernikeexpansion coefficient scaled by a scaling factor, the scaling factorcomprising a ratio of the second aperture dimension to the firstaperture dimension, the ratio raised to the power of a factor comprisinga radial degree of the coefficient.
 2. The method of claim 1, whereinthe aperture dimension comprises an aperture radius.
 3. The method ofclaim 1, wherein the optical system comprises an optical tissue of apatient, and the aperture dimension comprises a pupil dimension.
 4. Themethod of claim 3, wherein the pupil dimension comprises a pupil radius.5. The method of claim 1, wherein the first aperture dimension isgreater than the second aperture dimension.
 6. A method of determiningan optical surface model for an optical tissue system of an eye, themethod comprising: inputting a first optical data corresponding to theoptical tissue system of the eye, the first optical data comprising afirst set of normalized Zernike expansion coefficients corresponding toa first pupil radius of the eye; calculating a second optical datacorresponding to the optical tissue of the eye, the second optical datacomprising a second set of normalized Zernike expansion coefficientscorresponding to a second pupil radius of the eye, wherein the secondset of normalized Zernike expansion coefficient is calculated based onthe first set of normalized Zernike expansion coefficients scaled by ascaling factor, the scaling factor comprising a ratio of the secondaperture dimension to the first aperture dimension, the ratio raised tothe power of a factor comprising a radial degree of the coefficient; anddetermining the optical surface model based on the second optical data.7. The method of claim 6, wherein the first pupil radius of the eye isgreater than the second pupil radius of the eye.
 8. A system forcalculating a modified normalized Zernike expansion coefficient for anoptical system, the system comprising: means for inputting an originalnormalized Zernike expansion coefficient for the optical system, theoriginal normalized Zernike expansion coefficient associated with afirst aperture dimension; and means for calculating a modifiednormalized Zernike expansion coefficient for the optical system, themodified normalized Zernike expansion coefficient associated with asecond aperture dimension, wherein the modified normalized Zernikeexpansion coefficient is calculated based on the original normalizedZernike expansion coefficient scaled by a scaling factor, the scalingfactor comprising a ratio of the second aperture dimension to the firstaperture dimension, the ratio raised to the power of a factor comprisinga radial degree of the coefficient.
 9. The system of claim 8, whereinthe aperture dimension comprises an aperture radius.
 10. The system ofclaim 8, wherein the optical system comprises an optical tissue of apatient, and the aperture dimension comprises a pupil dimension.
 11. Thesystem of claim 10, wherein the pupil dimension comprises a pupilradius.
 12. The system of claim 8, wherein the first aperture dimensionis greater than the second aperture dimension.
 13. A method ofcalculating effective powers of an optical system, the methodcomprising: calculating a first effective power using a first Zernikeexpansion coefficient for the optical system, the first Zernikeexpansion coefficient associated with a first aperture dimension; andcalculating a second effective power using a second Zernike expansioncoefficient for the optical system, the second Zernike expansioncoefficient associated with a second aperture dimension, scaled relativeto the first Zernike expansion coefficient using a scaling factorcomprising a ratio of the second aperture dimension to the firstaperture dimension raised to a power of a factor comprising a radialdegree of the coefficient.
 14. The method of claim 13, wherein theoptical system comprises an optical tissue of a patient, and theaperture dimension comprises a pupil dimension.
 15. The method of claim14, wherein the pupil dimension comprises a pupil radius.
 16. A systemfor calculating effective power for an optical system, the systemcomprising: means for inputting a first Zernike expansion coefficientfor the optical system, the first Zernike expansion coefficientassociated with a first aperture dimension; and means for calculating asecond effective power using a second Zernike expansion coefficient forthe optical system, the second Zernike expansion coefficient associatedwith a second aperture dimension and scaled relative to the firstZernike expansion coefficient using a scaling factor comprising a ratioof the second aperture dimension to the first aperture dimension raisedto a power of a factor comprising a radial degree of the coefficient.17. The system of claim 16, wherein the optical system comprises anoptical tissue of a patient, and the aperture dimension comprises apupil dimension.
 18. The system of claim 17, wherein the pupil dimensioncomprises a pupil radius.